cas and its place in secondary school mathematics ... kemp cas in the... · without their love and...

CAS and its place in Secondary School

Mathematics Education

Andrew David Kemp

Dissertation presented as a partial requirement for the award of MSc (Mathematics Education of the University of Warwick)

Institute of Education University of Warwick

August 2008

2

Contents

Figures ................................................................................................................4

Abstract ...............................................................................................................7

Chapter 1 - Introduction to CAS ..........................................................................8

Chapter 2 - Literature Review ...........................................................................11

Chapter 3 - Designing an Intervention using CAS to teach a Year 8 Class ......35

Chapter 4 - Review the first implementation of the Activity and Refine the

Activity ...............................................................................................................58

Chapter 5 - Discussion of Results .....................................................................91

Chapter 6 - What belongs in a CAS Curriculum and how should we assess

it?.......................................................................................................................95

Bibliography.....................................................................................................125

3

Acknowledgments

Without several people this dissertation would never have happened. Firstly I

must thank Dave Pratt and Peter Johnston-Wilder who introduced me to the

academic side of Mathematics Education during my PGCE and MSc studies.

Dave and I both share a fascination with technology which has in no small part

shaped this dissertation. Peter deserves special thanks for working with me to

turn my original vague ideas into something that could be written up into this

dissertation.

Thanks also need to go to Andrea Forbes, Jenny Orton and Spencer Williams

at Texas Instruments for all their assistance in providing me with equipment and

inviting me to relevant conferences.

Finally the greatest thanks must go to my wife and children who have put up

with me reading and typing throughout most of our holidays over the last few

years. Without their love and support I know this would never have been

completed.

4

Figures

Figure 1 - CAS Calculator Screen .......................................................................9

Figure 2 - CAS Simplification ............................................................................18

Figure 3 - Using CAS as Scaffolding.................................................................27

Figure 4 - MEI June 2007 A-level Question ......................................................28

Figure 5 - Using the TI-Nspire CAS to integrate and differentiate.....................29


Figure 7 - Using the TI-Nspire CAS to solve implicit differentiation...................30


Figure 9 - Using the TI-Nspire CAS to solve a question about identites ...........31

Figure 10 - The Task - Part 1 ............................................................................44









Figure 19 - The Task - Part 10 ..........................................................................51






5




Figure 28 - TI-Nspire CAS Calculator................................................................60

Figure 29 - Triangle Numbers ...........................................................................66

Figure 30 - (1+x)n Binomial Expansions ............................................................68

Figure 31 - (a+b)n Binomial Expansions............................................................70

Figure 32 - (2x+1)n Coefficients.........................................................................72

Figure 33 - (2x+1)n Coefficients in terms of Pascal's Triangle ..........................73

Figure 34 - Student A's notes - Part 1 ...............................................................75




Figure 38 - Student B's notes - Part 1 ...............................................................79



Figure 41 - Student C's notes............................................................................82

Figure 42 - Student D's notes - Part 1 ...............................................................83

Figure 43 - Student D's notes - Part 2 ...............................................................84

Figure 44 - Student E's notes............................................................................85

Figure 45 - Poster 1...........................................................................................85

Figure 46 - Poster 2...........................................................................................87

Figure 47 - Poster 3...........................................................................................88

Figure 48 - Poster 4...........................................................................................89

Figure 49 - Mathematical Methods (CAS) Exam 1 - 2004 - Question 9 ..........106


6



Figure 53 - Danish STX December 2007 - Question 11..................................112

Figure 54 - AP Calculus BC Free Response Paper (Form B) 2006 -

Question 3 .......................................................................................................113

Figure 55 - MEI Core 3 June 2006 - Question 6 .............................................114


Figure 57 - Mathematical Methods (CAS) Exam 1 - 2005 - Question 27 ........116

Figure 58 - St Pölten, Austria 2001 Final Exam ..............................................117

Figure 59 - Mathematical Methods (CAS) Exam 2 - 2005 - Question 3 -

Part 1...............................................................................................................118

Figure 60 - Mathematical Methods (CAS) Exam 2 - 2005 - Question 3 -

Part 2...............................................................................................................119

Figure 61 - MEI Core 4 - January 2007...........................................................121

7

Abstract

In this essay I explore the current and potential role of Computer Algebra

Systems (CAS) within secondary education. I start by looking at the recent

research, paying particular attention to the current use of CAS within secondary

education and assessment worldwide. Much of the recent research into this

field is coming out of Australia’s Victorian Curriculum and Assessment Authority

who have been carrying out pilot studies since 2001.

Then I outline the creation of a CAS activity to be used with a class of year 8

students, and review what that might tell us about the potential use of CAS in

schools.

Finally I look at the potential shape of a CAS curriculum and the effect this

might have upon assessment, exploring how CAS would work within current

assessment mechanisms and suggesting possible alterations to existing

methods and alternative methods of assessment.

Chapter 1 – Introduction to CAS 8

Chapter 1 - Introduction to CAS

Computer Algebra Systems (CAS) have been around in various forms since the

early 1970’s. The first popular systems were Drive, Reduce and Macsyma.

Some of the currently available software packages in the field include

Mathematica, Maple, MathCAD, Derive and the open source project Maxima.

The first hand-held calculator capable of carrying out symbolic manipulation,

differentiation and limited symbolic integration was Hewlett-Packard’s HP-28,

which was released in 1987, and which was followed by the HP-28S a few

years later and the HP-48SX in 1990. Then in 1995 Texas Instruments

released the TI-92, which has an advanced CAS system based upon Derive.

Casio finally introduced a CAS capable calculator in the form of the FX-9970G

in 1998, followed by the FX Algebra 2.0 in 1999.

More recently Casio have produced their ClassPad line and Texas Instruments

have released the TI-Npsire CAS calculator. These retail for around £150 each

and are capable of carrying out extensive symbolic calculations, including

solving systems of equations, indefinite integration and differentiation and

automatic simplification of algebraic statements.

Computer Algebra Systems allow for a huge range of additional features above

and beyond those available in a standard numerical calculation environment.

Within a CAS environment it is possible to simplify algebra and manipulate it in

a very similar way to that in which a standard calculator enables the


manipulation of numbers. CAS systems can factorize and expand expressions,

solve equations exactly, simplify algebraic fractions and rewrite trigonometric

functions. They can carry out full and partial differentiation, solve systems of

linear and some non-linear equations, take some limits, and carry out many

indefinite and definite integrals (including multidimensional integrals). They can

also carry out Series and Product calculations, and manipulate Matrices. Some

CAS systems also allow you to evaluate solutions to very high levels of

accuracy using bignum arithmetic which allows calculations on integers or

rational numbers to be carried out to an arbitrary number of decimal places,

limited only by the amount of available memory. Many also include graph

plotting facilities and high level programming languages which allow users to

implement their own algorithms. Examples of the kind of things that CAS

systems can do can be seen in the image below:

Figure 1 - CAS Calculator Screen

The focus of my work is on the use of CAS with KS3 & KS4 students, who

wouldn’t normally have access to this technology. I am most interested in a


very limited set of functions made available by CAS. My use of CAS will mainly

focus on using CAS to carry out skills usually acquired by students during KS3

& KS4: expanding and factorizing brackets, solving linear, quadratic and

simultaneous equations and simplifying expressions. Using a Computer

Algebra System to carry out these tasks I hope will enable them to work at a

higher mathematical level then they would normally be expected to be able to

achieve. I also hope that the students will be able to acquire some of these

skills for themselves through the Constructionist (Papert 1980, 1993) approach

of the task I will explore later.

Chapter 2 – Literature Review 11

Chapter 2 - Literature Review

The use of CAS systems in schools has from the outset been very

controversial. In many ways the concerns mirror those experienced when the

first ‘4 function’ calculators were introduced (Reiling & Boardman 1979 pp. 292-

294). The fear with both the introduction of simple calculators and with the use

of CAS systems is that students will lose the ability to carry out these skills

manually and thus will have lost something that is an essential part of

mathematics (Gardiner 1995, Taylor 1995 p83).

However one of the questions that must be answered is what are these

essential skills? What is really an essential part of mathematics? When

calculators were introduced that could carry out square roots it was felt by many

that if we didn’t continue to teach the algorithm (something which almost no

current students will have met) for finding square roots then students would not

really understand what a square root was (French 1998 p65-66). However the

fact that this process is no longer taught in schools implies that for whatever

reason the learning of this process was deemed to be non-essential.

An interesting debate took place within the Association of Teachers of

Mathematics journal MicroMath about what were the essential manual methods

that must be maintained with the availability of CAS calculators. The article by

Herget, Heugl, Kutzler and Lehmann (2000) suggests that the use of CAS

Calculators will soon become widespread and common place:


“[CAS Calculators] will be used as a matter of course, much as we use scientific

(in some countries graphic) calculators today, Using a calculator for

differentiating will be as common as using it to evaluate

cos(1.3786) or ” Herget et al (2000 p10).

The writers present their exploration in the form of three pots within which they

place exam style questions. The three pots are -T (Technology Free Questions

– allowing no access to any calculating devices), +T (Technology Assisted

Questions – in which students are allowed access to powerful calculators or

CAS devices) and ?T. This pot represents their doubts over the boundary

between technology enabled and technology free questions.

The examples they suggest for a non-calculator exam include things like

“Calculate ” or “Estimate ” or “Factorize 15”. On the other hand they

suggest that with a CAS calculator they should be able to answer questions

more like “Compute ” or “simplify ” or “find the factors of 30”.

(Herget et al 2000 p12)

The basic rule they are suggesting with the use of CAS is:

“Elementary calculations (such as factoring of an integer with only two factors

e.g. 15) are an indispensable skill (therefore these skills belong to -T). On the

other hand, calculations requiring repeated applications of elementary

calculations (such as factoring of an integer with three or more factors, e.g. 30)

may be delegated to a calculator.” Herget et al (2000 p12)


The idea that a calculator should be used to find the factors of 30 was not a

popular suggestion with Gardiner (2001). In his response to Herget et al he laid

heavy criticism on the idea of deciding the difficulty of the question based on the

number of factors.

“This is patent nonsense - pedagogically, didactically and

mathematically. Are English pupils alone in thinking that 91 is

prime? How can anyone know in advance whether an integer is the

product of two primes (and so according to the authors’ criterion, that

it should be factored by hand)? Is it not obvious that 30 is more

accessible than 91, and that factoring (and much

larger numbers) by hand plays a significant role in helping pupils

understand the workings (and the consequences) of prime

factorisation?” Gardiner (2001 p.8)

Whilst I agree with Gardiner to an extent I have some sympathy with the original

intent of Herget et al. What I felt Herget et al were trying to suggest is that

students should be able to understand the concept of factorization and be able

to carry it out without aid from calculators. However, students should not be

required to spend undue time carrying out the mechanical process when this

could be done for them by their calculator.

Next looking at the field of Fractions, Herget et al suggest that questions like

“Simplify ” or “Simplify ” belong to the non-calculator pot whereas

more sophisticated questions like “Simplify ” and “Simplify


” belong in the CAS environment. Again they are, in my opinion,

taking the perspective of testing understanding of the concept, without requiring

students to carry out the task by hand when it involves demonstrating the same

skill repeatedly, as can be seen by the two examples shown for the CAS

environment.

Next they explore the use of brackets and equations. Again, they seem to

approach this field from the suggestion that testing basic understanding with

simple examples is expected without access to a calculator but repeated

applications or multistep problems are predominately reserved for the CAS

environment.

In the field of Quadratic Equations the suggestion is that testing of the

Quadratic Formula should no longer be considered a requirement. As, from my

experience, many students cannot use the result properly and those that can

are not really sure why it works, this would seem a sensible omission for most

students. There is however a case for retaining some concepts for just the

most exceptional students, for whom the time taken to cover the curriculum

content is not such a limiting factor.

Finally in the field of Differentiation Herget et al (2000) limit pencil and paper

methods to carrying out routine simple differentials of polynomials or basic

results such as the differential of sin(x), or ln(x). This means that anything

that would require use of the Chain Rule or the Product Rule would be reserved

for CAS questions.


Both Gardiner (2001) and Monaghan (2001), in their responses to Herget et al,

raise concerns about this method of exploring what CAS should be used for.

Monaghan indicates that there are really two issues in question: What are the

“indispensable manual calculation skills in a CAS environment”? and What

should “assessment with and without CAS technology” look like? (Monaghan

2001 p.9). He claims that the two areas, whilst related, are really very different.

I will return to the issue of assessment within a CAS environment later in this

section, and again when I consider the shape of a CAS curriculum in chapter 6.

So returning to the question of what skills are indispensable in a CAS

environment we must realize there are really two questions here as well: ‘What

skills do we want students to use without access to a CAS calculator?’ and

‘What skills will students need in order to use a CAS calculator effectively?’

I first deal with the second issue, that of skills required to use CAS effectively,

before returning later in this chapter to the question of what manual skills we

require students to still be able to complete.

Lagrange (1999) explores some ideas related to students’ initial interactions

with CAS. He does this by considering how students are likely to use a CAS

calculator at first. Looking at the example of simple division, an area that

students already have some preconceptions about what the answer should look

like, he looks at what happens when the student carries out the following

calculation:


“When a beginner uses his/her new TI-92 to do a division, like 34 divided

into 14, s/he keys in like on an ordinary

calculator and s/he is very surprised when the TI-92 answer with 17/7”

Lagrange (1999 p66.)

Here the student is forced to consider the difference between the

“approximations of everyday practice” (Lagrange 1999 p66) and the exact

values that result from mathematical treatment of numbers. This difference is

something that more teachers have to deal with since Casio produced their

FX83ES scientific calculator

(http://www.casio.co.uk/Products/Calculators/Scientific%20Calculators/FX83ES

viewed 13th Feb 2008), which includes what they describe as “Natural

Textbook Display”. Carrying out the same calculation as Lagrange uses in his

example results in the same solution on these now common scientific

calculators.

The issue here for students is to appreciate the difference between an exact

fractional answer and the equivalent decimal approximation. Whilst it is still

possible, with each of these calculators, to obtain the decimal approximation,

the default output of each is exact. The same is true when dealing with surds

and, in the case of a CAS calculator, exact values are also given for

trigonometric values, where these are possible.

I think the emphasis on exact values is something that mathematics education

in England has lost as a result of the emphasis on calculator usage over the


past 20-30 years in schools. Gardiner (1995), talking about the effect of the

calculator, says:

“Some numbers - integers, powers of two, primes, unit fractions, pi, e,

, and so on - are more interesting than others.

The conviction that some numbers are more interesting than others has

to be developed separately from, and preferably before, the experience

of being subjected to calculators which present all outputs in the same

decimal format. Instead of learning that some numbers are more

interesting than others, those whose experience of numbers and

arithmetic is tied to the calculator see all numbers as equally

uninteresting decimals.” Gardiner (1995 p532 - emphasis original)

Whilst these complaints are true of the standard calculator, they do not hold

true for the CAS enabled calculator. With a CAS calculator, students are,

wherever possible, given answers in exact form drawing their attention, and

hopefully their interest, to these special numbers.

Lagrange continues on to look at the way in which the TI-92’s automatic

simplification can cause confusion to students.


Figure 2 - CAS Simplification

Here two mathematically equivalent statements have been simplified in quite

different ways. It is important that students understand that these are really the

same. Otherwise, there is a danger that students might construct their own

incorrect interpretation that leads them to conclude that the order of these

operations changes the meaning. The students must “consciously learn to use

the items of the algebra menu (Factor, Expand, ComDenom), to decide whether

expressions are equivalent as well as anticipate the output of a given

transformation on a given expression.” (Lagrange 1999 p67)

Lagrange’s work was focused on equivalent expressions. In one of the tasks

students were given an expression, G , and asked to

identify which of three statements are equivalent to G.

The other expressions they were given were: H ,

I , and J .

“Students had no difficulty seeing that I was the opposite of G... In

contrast, showing that G and H are equivalent, is not straightforward: in


the screen B, the function for the reduction of a sum of rational

expressions has been used to reduce a sub-expression. Factor was the

appropriate function to obtain J from G.” (Lagrange 1999 p68)

This use of CAS rather than traditional paper and pencil methods forces

students to look at the different ways in which the same mathematical

expressions can be represented in different forms. As Lagrange indicates the:

“conscious use of the algebraic capabilities of the TI-92 may help

students to focus on the most suitable form for a given task, whereas

paper and pencil schemes focus on the rules of transformation. One

may reasonably think that the joint development of the TI-92 and

paper/pencil schemes is able to give an understanding of the

equivalence of expression. This is an example of how paper and

pencil and TI-92 practices are to be thought complementary in

teaching, rather than opposed.” (Lagrange 1999 p68)

This implies that there may be something to be considered about a hybrid

approach of using CAS in concert with paper and pencil methods. It also lends

some weight to the idea that CAS calculators could be used effectively in the

teaching of Mathematics whilst still not being available for use within

examinations.

Heid & Edwards (2001) talk about the fact that over the past 20 years

mathematics education has been “characterized by calls for (and movement

toward) fundamental changes in the nature and purpose of school algebra”

(Heid & Edwards 2001 p129). The changes that Heid & Edwards allude to here


are a move towards a more multi-representational approach. The multi-

representational approach revolves around the idea of seeing the same

mathematical object in various forms - graphs, tables of values & symbolic

rules.

The increase in classroom accessible technology “has enabled teachers to

provide their students with a richer approach to mathematics” (Heid & Edwards

2001 p129). This approach moves away from the primarily symbolic approach

to algebra which typified our pre-technology approach. The focus is on

“symbolic reasoning instead of primarily on symbolic manipulation” (Heid &

Edwards 2001 p129).

Within the CAS enabled environment “Symbol sense, symbolic reasoning, and

symbolic disposition assume greater importance in a world in which computer-

based algebra exists” (Heid & Edwards 2001 p129). CAS offers opportunities

for students to develop their symbolic understanding through several avenues.

They can “outsource” (p129) routine work to the CAS, thus enabling them to

focus on the holistic problem without becoming entrenched in the detail; to

perceive some of the multiple representations mentioned previously, including

multiple equivalent symbolic representations, and interpret the different

information they provide; to “bridge the gap” between the concrete examples

and the abstract generalization; and to examine “symbolic patterns (more

concretely)” (Heid & Edwards 2001 p129).

The hope, as Heid, Choate, Sheets & Zbiek (1995) suggest, is that:


“Students too often leave their algebra experience with a modicum of

ability to produce equivalent forms but very little understanding of the

meaning of that equivalence. In a technological world in which

students have access to computer-algebra utilities… the importance

of producing equivalent forms no longer overshadows the importance

of understanding what the equivalent expressions mean.” (Heid,

Choate, Sheets & Zbiek 1995 p 127)

This for me is the crux of my hope for the use of CAS in secondary schools. My

hope, much like Heid et al, is that the use of CAS will enable students to

transcend the routine nature of symbolic manipulation and to focus better on the

bigger picture.

Heid et al (2001) also consider the possible roles that CAS might play within the

classroom and offer four possible alternatives. The first possibility is to consider

and use CAS as a “producer of symbolic results”. This would enable the focus

within the curriculum to move from a skills based curriculum to a curriculum

which emphasizes the “development of mathematical concepts or

understanding of applications over the acquisition of paper and pencil

manipulation skills” (Heid et al 2001 p130).

The second possibility for the use of CAS is to enable students to learn

symbolic manipulation skills. For example, students can use CAS to help them

solve an equation like . But, rather than using the SOLVE command,

they can manipulate the equation, and if they make mistakes, they can use the

“delete key… to clear off this step and try again” (Heid et al 2001 p 131). Heid


et al describes the process of solving this equation using the TI-89. One of the

positive features of this method is that calculator will always correctly carry out

the operation the student has suggested. This means that students cannot

carry out invalid steps which lead to the correct answer. This example

illustrates the use of a CAS system as a pedagogical tool. Within the CAS

environment students are assisted in “constructing sound conceptual

understandings that underlie symbolic manipulation” (Heid et al p131).

A third possibility for the use of CAS in the classroom is to aid students in the

generation of multiple examples from which they can engage in the process of

pattern spotting. Nice examples of this include things like exploring the

Binomial Expansion as an investigational activity or exploring questions like

for different values of n.

The fourth possible use of CAS lies in exploring solutions to general or abstract

problems. The example that Heid et al refers to is the following problem: “Find

the roots of any quadratic equation”. CAS enables them to use the SOLVE

command with dummy variables. Issuing the command “Solve (ax^2+bx+c =

0,x)” would give a variant upon the traditional Quadratic Formula. Taken as a

starting point for some investigational work this may lead into a much richer

understanding of the Quadratic Formula than either presenting it as a fact to

learn or giving the proof, which students, in my experience, struggle to follow.

The four possible uses I have just discussed can be described using the ‘Black

Box’ and ‘White Box’ metaphors which were espoused by Buchberger (1989).

Buchberger questions whether we should continue to teach integration


techniques when CAS can do it for us. For Buchberger, integration isn’t the

issue, but rather how we react to the teaching of an area of mathematics which

has been trivialized. Buchberger says that an area of mathematics can

described as ‘trivialized’ “as soon as there is a (feasible, efficient, tractable)

algorithm that can solve any instance of a problem in this area” (Buchberger

1989 p1).

Buchberger starts by exploring the possible extremes, first that we no longer

teach integration and that “students should not be tortured with it any more”

(Buchberger 1989 p2). He counters this point of view by explaining that

integration techniques are not only taught so they can be applied to more

complex problems but “mainly, because, by teaching the areas, students gain

mathematical insight and learn important general mathematical problem solving

techniques” (Buchberger 1989 p3).

Next Buchberger explorers the antithesis of this, that is the idea that we simply

ignore CAS and ban it from education. This seems to have been the

mainstream reaction to CAS thus far, certainly within the English education

system where the use of CAS is minimal at best. The argument tends to go

along the lines of CAS will “spoil students similarly as pocket calculators have

spoiled kids” (Buchberger 1989 p3).

The counter argument here can take the form of the question, “why are we

bothering students with learning mathematics that is now trivialized when the

time could be used to move on to more advanced mathematics?” Buchberger

goes on to suggest that, in fact, CAS could “open the chance to drastically


expand the number and in-depth-treatment of subjects covered in a typical

math curriculum because significant time could be saved during undergraduate

education.” (Buchberger 1989 pp. 3-4)

So having dealt with and discounted the two extremes Buchberger (1989 p4)

introduces the didactical principle of ‘white box’ vs ‘black box’ for using

Symbolic Computation Software in Math Education. The idea of a ‘black box’ is

used to refer to a device, system or object which is viewed solely in terms of its

inputs and outputs. The concept of a ‘white box’ however is one in which the

inner working or logic of the device, system or object is available for all to see.

Buchberger argues that, where the concepts are new to the student, the use of

CAS as “black boxes would be a disaster” (Buchberger 1989 p4). However,

once the topic has been thoroughly studied “students should be allowed and

encouraged to use the respective algorithms available in the symbolic software

systems” (Buchberger 1989 p4)

Buchberger (1989 p6) suggests that the white box/black box process should be

used as a recursive procedure. Using Integration as an example, students

should be encouraged to study and understand the concept of Integration.

Once students thoroughly understand the concept of Integration, a CAS device

can be used. This means that, when the student moves on to studying

differential equations, they can use CAS to carry out routine integration

techniques. Then, once differential equations are fully understood, the CAS

device can be used to solve the differential equations and this can then be used

as a tool within the next topic.


However, to imply that CAS should only be used as a tool once the skills have

already been acquired is not a viewpoint shared by everyone. Heugl (1997

p34) implies that using CAS as a ‘black box’ might be appropriate for students

to develop for themselves the techniques demonstrated by the CAS, or as

Heugl puts it, for students to reach the point where they can say “we are able to

do what the CAS can do” (Heugl 1997 p34.)

Lagrange (1999) suggests however that this view is over-simplistic. In his

opinion, using CAS as a black box will only enable students to discover

'symbolic entities', by which he means that students can learn the techniques

but that learning theories and concepts would require a wider range of

strategies. He does however agree that due to CAS's "focus on symbolic

aspects of concepts, it could be useful for teaching symbolic rules" (Lagrange

1999 p74). Lagrange has in mind here the third possibility, presented by Heid

et al (2001), whereby students "could consider several examples… and then

learn to do those calculations by themselves" (Lagrange 1999 p.74). Lagrange

however feels that implementing this type of process will not be simple. Using

the work of Pozzi (1994), he states the need for students using CAS to have

reached a sufficient level of algebraic maturity. On the other hand he does

claim that "computer algebra systems can support students to make sense of

their algebraic generalization” but that this is "only likely to be achieved if

[students] use the computer to explore and verify their conclusions and not

simply as a symbolic calculator" (Pozzi 1994 cited in Lagrange 1999 p 75).


Kutzler (1999) proposes the use of CAS as a form of scaffolding to support

students. Using the metaphor of building a house, Kutzler draws attention to

the fact that, when teaching mathematics in schools, we “simply don’t have

enough time for waiting until all students have completed all previous storeys”

(Kutzler 1999 p7). As teachers, the curriculum forces us to continue from topic

to topic, “independent of the progress of individual students” (Kutzler 1999 p7).

So the question Kutzler is attempting to explore is how a student can build a

storey on top of an incomplete one.

Kutzler suggests that the CAS might be able to act as a form of scaffolding

enabling the student to learn “the higher-level skill… [whilst] the calculator

solves all the sub-problems that require the lower-level skill” (Kutzler 1999 p8)

Kutzler goes on to demonstrate how this concept of CAS-based scaffolding

might work when solving a system of equations using Gaussian elimination. He

demonstrates how a CAS system can be used to carry out the multiplication of

the equations and then the subsequent addition or subtraction. The advantage

of using CAS is that it eliminates the common experience we see in schools

whereby:

“some students choose the right linear combination, but, due to a

calculation error, the variable does not disappear. Other students

choose a wrong linear combination, but, again due to a calculation

error, the variable disappears… For both groups of students their

weakness in algebraic simplification is a stumbling block for

successfully learning the basic technique of Gaussian elimination.


Exactly those students lag behind more and more the ‘higher’ they

get in the house of mathematics” (Kutzler 1999 p8)

Kutzler proposes a 3 stage approach, whereby if we are trying to develop skill B

(in our example Gaussian elimination) and this requires skill A (in our example

Algebraic manipulation) then we start with the first step of teaching and

practising skill A. The second step is teaching and practising skill B, whilst

using a CAS calculator to deal with any sub-problems that require skill A,

allowing the students to fully concentrate on learning skill B. The final step is to

combine these two skills together with no support from technology:

Figure 3 – Using CAS as Scaffolding

Kutzler claims that this use of technology helps break down the learning

process into smaller more manageable pieces, which students find easier to

keep track of without “getting lost (or screwed up) in details such as

simplification” (Kutzler 1999 p9)

For this reason, Kutzler proposes that the use of CAS calculators “should be

introduced as a pedagogical tool independently of any changes to the

curriculum or the assessment scheme.” (Kutzler 1999 p9)

I will now return to the question of assessment, which I first looked at as a

method of defining what I felt was essential paper and pencil skills that we


should retain. Now I will look at some of the implications of using CAS for the

way we assess students’ abilities.

Taylor (1995) points out that CAS calculators “carry out much of the routine

algebra and calculus currently examined at A-level” (Taylor 1995 p 74),

including factorization, solving systems of equations, expressing partial

fractions, differentiating and integrating expressions, summing series, and

calculations involving complex numbers. He then draws attention to several

questions for which a CAS calculator would be of direct benefit.

The following questions from June 2007 MEI A-level Mathematics papers, show

that Taylor’s observation that CAS calculators could carry out much of the

algebra and calculus on the A-level papers in 1995 still remains true now:

Figure 4 - MEI June 2007 A-level Question

These first two questions would be completely trivialised if the use of a CAS

system was introduced. The student simply needs to enter the differential or

integral and the CAS system will give them the answer. An example of how this

might work is shown below completed with the TI-Nspire CAS:


Figure 5 - Using the TI-Nspire CAS to integrate and differentiate

Next we will look at another question which can be solved much more simply

with use of CAS. This one however is a little more complicated as it requires

the use of implicit differentiation.


Similarly, this second question can be completed simply with the use of CAS.

This time it requires the use of another function impDif to carry out the implicit

differentiation. Here syntax becomes more important, as the order in which you

list the variables in the impDif function determines whether you calculate or

. For the second part we can combine this with the Solve command to find

when this is zero.


Figure 7 - Using the TI-Nspire CAS to solve implicit differentiation

Finally we will look at an example of how CAS can be used with identities.


This final question shows how CAS can be used to support an algebraic

question without necessarily carrying out the whole calculation. One approach

is to simply enter the result as shown (replacing the identically equal sign with

an equals sign) and then manually equate the coefficients. Alternatively you

can rearrange the equation into , and then ask

the CAS to carry out a polynomial division. The first approach is to use CAS to

carry out the expansion, but this leaves the student to complete the task by

hand. The latter approach involves the student reformatting the question into a

format that the CAS can more easily solve.


Figure 9 - Using the TI-Nspire CAS to solve a question about identities

So, as can be seen, with the aid of CAS some of the questions are greatly

simplified but others, like the first two, become trivial, as once entered onto the

CAS calculator the student would only need to write down the answer given.

In considering how examinations may need to adjust, we can begin by exploring

what other countries have done with respect to CAS. Work on the use of CAS

has focused on a small number of countries. The use of CAS in the US College

Board’s AP Calculus exams has been allowed since 1995 and is now well

established; the French Baccalaureate Générale Mathematics has allowed the

use of CAS since 1999; the Danish Bacclaureat Mathematics component has

allowed their use since 1997. Within Austria and Switzerland, teachers set their

own examinations and may allow students access to CAS calculators if they so

wish. The International Baccalaureate Organisation (IBO) started a pilot CAS

Higher Level programme in 2004. New Zealand is actively exploring a CAS

enabled curriculum. Finally Australia ran a pilot Mathematical Methods (CAS)


examination from 2001 - 2006, which has now been rolled out to all schools in

the Victorian Curriculum and Assessments Authority. (Evans et al 2005 p330)

A review of the thoughts of a working group of the, then, National Council for

Educational Technology (NCET), as reported by Monaghan (2000 pp. 383-384),

described four broad styles of A-level questions which offered no discernible

advantage to CAS users.

These included questions which involved Graphical interpretation, specifically

questions where there is no advantage to using a graphical calculator, such as:

Complete this partial sketch of f(x) given that f is an odd function. The second

form were those which involved Algebraic forms not suited to straightforward

CAS manipulation; examples of these include: “express in

the form ”. The third form were those which involved Computational

interpretation. These tended to be questions where the calculations were fairly

trivial but the interpretation was the challenge. For example, given the position

vectors of three points, give the vector equations of the lines that pass through

any two points. The final type of question consisted of those that involved

modelling skills.

The NCET working group felt that, in most cases where the questions offered

possible advantages to CAS users, it was possible to alter the questions so that

the advantage was at least minimised. They found in most cases one of the

following techniques would help balance out the advantage that CAS might

provide. The first technique was to Introduce Parameters. For example,


instead of ‘factorize the following cubic’, one could ask the student to

‘Determine the value of a for which has a factor of (x-1)’.

The second approach is to Give the Result in the Question. It has become

increasingly common for a question to be phrased in the form “Show that…”.

This also has the advantage of allowing students to access later parts of the

question even if they are unable to show the desired result.

The third approach is to require Specific Methods. Instead of asking students to

simply integrate a function, ask them to integrate by parts or by using a

particular substitution.

The other possible approach is to situate the question in a context and give a

“greater emphasis to modelling and interpretation” (Monaghan 2000 p.385)

These ideas address the way in which we can adapt our current syllabus and

make it more CAS-compatible, without having to address any major changes to

what and how we teach, which is the approach that has been undertaken by

most CAS-enabled examinations. The emphasis here is on removing the

advantage of CAS rather than embracing it and exploring the potential shape of

maths education if CAS were available. Monaghan notes that the UK exam

boards reacted in a similar way to the availability of Graphical Calculators when

they “changed the style of some questions in the light of calculator

developments but … did not develop syllabuses which required a graphical

calculator.” (Monaghan 2000 p382)


In the final chapter I will consider what changes might be appropriate to the

syllabus in the light of the availability of CAS within our classrooms as well as in

our examinations.

Chapter 3 - Designing an Intervention using CAS to teach a Year 8 Class 35

Chapter 3 - Designing an Intervention using CAS

to teach a Year 8 Class

My work using CAS with students will be in two parts. The first section will be

introductory and will involve a series of simple directed tasks to familiarise the

students with the CAS system. These tasks will be highly focused and will

concentrate on the terminology and syntax of the CAS system which will be

new to the students. This will form the backdrop to my work with CAS and

therefore will not form a significant part of my evaluation. Following on from

this, the students will use the CAS environment to explore the Binomial

Expansion. I have chosen the Binomial Expansion as my topic as it is usually

not studied until much later within the mathematics curriculum (typically at A-

level); however, I believe that through the use of CAS the topic becomes

accessible to much younger students with only an elementary understanding of

algebra.

During the planning of the Binomial Expansion task I explored a large collection

of CAS tasks from a variety of sources to explore how other people have used

CAS within secondary school teaching, to identify any common threads and to

evaluate the potential of a CAS system.

In exploring the range of available tasks, I began by looking at exemplar tasks

provided by Texas Instruments to accompany the TI-Npsire CAS Calculator

(http://compasstech.com.au/TNSINTRO/TI-NspireCD/HTML_files/activities.html

and http://compasstech.com.au/TNSINTRO/TI-


NspireCD/mystuff/showcase.html, accessed 4th April 2008) and the materials

provided by Casio to accompany their ClassPad 300 Calculator

(http://www.casioeducation.com/activities/lesson_calc/2AF02BE1-A8B2-44E2-

AEEA-72A561F21A36! accessed 4th April 2008).

Both of these resources provide a series of self-contained lessons which cover

various mathematical topics from geometry to calculus. Most take the form of a

PDF lesson outline and then an activity file, which should be copied to the

calculator, and which contains either questions or an environment within which

students can explore the topic.

These resources present several different ways in which CAS can be used

within school teaching. Some of the activities use CAS as scaffolding to

support the student by indicating at each stage if the step the student has taken

is valid. An example of this type of activity is presented by Steven Arnold

(http://compasstech.com.au/TNSINTRO/TI-

NspireCD/mystuff/dynamic_algebra.html! viewed 4th April 2008). Within this

activity, Arnold demonstrates manually solving equations, and using the CAS

environment to check the working at each stage of the process. This is a

demonstration of what Arnold (2008 p.3) describes as “Dynamic Algebra”.

Within this he explains that:

“The real challenge in using CAS effectively for teaching and learning

(especially with younger students) is to not let the tool do all the

work! My current preferred model offers students scaffolding without

doing the work for them – just checking their work as they go!”

(Arnold 2008 p.4)


Other alternatives are to use the CAS system as a means of avoiding the

algebra in a question, in much the same way as scientific calculators are used

to avoid the arithmetic complexity involved in some problems. A nice example

of this is an investigation into the best position from which to kick a Rugby ball

for a conversion (http://compasstech.com.au/TNSINTRO/TI-

NspireCD/Exemplary_Activities_PDF/Act9_PlayingRugby.pdf, accessed 4th

April 2008). Within this activity, the students reach the point where they need to

differentiate an inverse tangent function, a skill beyond the ability of students for

whom the activity is intended. It is assumed that students have an

understanding of elementary differentiation and know that the turning points

occur where the derivative is zero. Students can then use the CAS

environment to find the actual derivative, hence allowing students meaningfully

to describe the solution to a problem that would otherwise have been beyond

their mathematical repertoire.

A third use of CAS is to allow for the construction of a mathematical

environment within which students are given the opportunity to explore a

mathematical situation in such a way that they can discover some mathematical

property for themselves. This often takes the form of pattern spotting, where

the CAS system is used to generate examples; these examples are then

analysed for patterns. These patterns can then be used to generate an

hypothesis, which can then be further tested and refined using the CAS

environment. A good example of such an activity can be found in the “Algebra

Tools” activity (http://compasstech.com.au/TNSINTRO/TI-

NspireCD/Exemplary_Activities_PDF/Act7_AlgebraTools.pdf, accessed 4th


April 2008). Within this activity the student explores the effect that changing the

value of a has upon the function . The activity enables the

student to explore the question in both an algebraic and graphical form and

encourages the student to form and test their hypotheses.

When thinking about what approach I wished to take in my task design, I had to

consider how my philosophy of education might influence the problem. My

general approach to mathematics education is Constructionist and is heavily

influenced by the work of Seymour Papert (1980, 1991).

The key Constructionist concept I will be using as the basis of my task design is

that of a ‘microworld’ which was made famous by Papert (1980). In his seminal

book “Mindstorms”, he describes microworlds as:

“incubators for knowledge... First, relate what is new and to be learned to

something you already know. Second, take what is new and make it your own:

Make something new with it, play with it, build with it.” (Papert 1980 p.120)

The concept of a microworld has over time become very inclusive to the point

whereby “microworlds are frequently described in the literature in terms of a

collection of software tools.” (Hoyles & Noss 1990 p.415)

However there should be more to a microworld than it simply being a collection

of software tools. A microworld should be “a set of activities in a computational

setting, designed to be ‘rich’ and ‘dense’ as far as the predetermined

mathematical domain is concerned.” (Hoyles & Noss 1992)


My intention is to design a CAS microworld, within which students can explore

the concept of the Binomial Expansion. Like many microworlds (LOGO for

example), this microworld will have the potential to achieve much more, but by

focusing in on a few commands it is possible to create a focused environment in

which students can explore, conjecture and test their ideas about the Binomial

Expansion.

The idea is that the students will use the facilities of the CAS environment as a

method of gathering data so that the problem of describing the binomial

expansion can be looked at as a pattern spotting exercise. By turning the

problem into one of examining patterns it becomes much more accessible to

younger students who do not have the algebraic sophistication to approach the

problem from an analytical perspective.

The design follows the investigational approach presented by Polya (1957 p.5-

6), in which he outlines the four phases of problem solving. The first phase is

“to understand the problem” (p.5 emphasis original). This is the phase within

which the students must come to understand clearly what they must do.

In the second phase, students “make a plan” (p.5 emphasis original). This is

the point at which students try to identify how the parts of the problem might be

connected and how the variables link to the data.

The third phase is where the students “carry out [their] plan” (p.6 emphasis

original) and finally in the fourth phase the students “look back at the complete


solution” (p.6 emphasis original), checking their results to make sure they fit the

original problem.

I will be working with an above average ability group of 23 boys (a middle band

group from a selective boys school), who have little experience of working on

open-ended investigational work like this. Therefore, to aid the process I will

provide them with a framework based on the above.

The task itself will be carried out using the Texas Instruments TI-Npsire CAS

calculator. The students will have access to the calculators during lessons for

around 3 hours of teaching but will not take them home.

My hope is that the use of a computerised system will enable them to feel more

free to experiment than they might if they were working on paper. Falbel

expresses it well when he explains the effect that the word processor has had

upon writing:

“The computer can turn a piece of text into a fluid, plastic substance

that can be edited and manipulated at the touch of a few buttons.

Revision is no longer the arduous task it once was, and ease of

editing can liberate people to be more expressive and free in their

writing” (Falbel 1992, p33).

My hope is that the CAS environment will give students a similar freedom to

explore and revise their ideas without fear of making mistakes. However, it is

important to appreciate that the scope of this project is quite small and so it is

unlikely that many of the students involved in this study will reach the level of


skill in using CAS required for them to feel that sense of liberation. A further

long-term study would be required to look at students’ attitudes before and after

an extended period of use with CAS to verify my hypothesis that the use of CAS

can achieve this liberating effect within the mathematics classroom. My hope

however is that this study will demonstrate that the use of CAS in the classroom

can allow for the possible restructuring of the current mathematics curriculum.

The Problem the students will be asked to investigate will be to describe the

expansion of . Following the approach of Polya (1957) as outlined

above the task begins by exploring the problem from an intuitive point of view,

playing with a few simple results, establishing the effect that changing a, b and

n has upon the general problem. The students at this stage should be realising

that as the problem stands it has too many variables, and is too complex. As a

result the students will be encouraged to “try to solve first some related

problem” (Polya 1957 p.10). The related problem the students will be

encouraged to explore is the expansion of as this restricts the problem

to just one variable, n.

The students will then be encouraged to gather some data; this will be done by

using the calculator to expand for small values of n and then recording

the results produced by the calculator. Again, to make the problem more

accessible, the students will be encouraged to explore just the coefficients of

the expansion, recording these for the first 4 or 5 values of n.

When the students have gathered some data and have some results, they can

continue to explore the data until they feel they have a plan in place which


would allow them to answer the problem for a higher value of n (i.e. State the

coefficients of ). At this stage they can then move onto phase four and

examine the solution they have by comparing it to the solution provided by the

CAS calculator.

Having solved this simplified version of the problem, the students will then be

encouraged go back and see if their insight to the simplified problem of the

coefficients might now help them answer the more complex problem they

started with of finding the expansion of (rather than just the

coefficients), again following the strategy outlined above.

Then, once they have successfully been able to describe this expansion they

can return to the original problem of the expansion of , and work

towards being able to describe the expansion of something like . Once

they have gathered some data they can again attempt to form a plan of attack

for this problem, which they can then verify by comparing with the results

obtained by the CAS calculator.

I will use a similar model to that used by the Texas Instruments exemplar

materials (http://compasstech.com.au/TNSINTRO/TI-

NspireCD/HTML_files/activities.html, accessed 4th April 2008) of producing a

calculator document for the students which will focus their use of the calculator

and give some structure to their investigation.

These calculator documents allow the teacher to design an environment which

can include multiple linked representations (for further details, see http://www.ti-


nspire.com/tools/nspire/features/multi_rep.html, accessed 4th April 2008) of the

same object. For example, you could solve 2 simultaneous equations by

entering and manipulating them in the ‘Calculator View’ but, at the same time,

have the graphs of the two functions visible and have a table of values

calculated. Similarly you could geometrically construct a circle, which could be

manipulated to change the size to produce data points of radius against

circumference or area; then find a line of best fit to describe the results and

hence ‘discover’ the value of pi.

I think that using documents like these as microworlds will play a significant role

in the future use of technology in teaching mathematics.

I will now work through the task outline as it was presented to the students. I

began by creating a multi-paged TI-Npsire Document. Some of the pages

contain only text for the students to read or questions for them to complete on

paper. Others contain both instructions and a calculation window. The first

page, as can be seen below, introduces the problem in a very general sense

and reminds students how to move between pages within the document. It

draws particular attention to the generality of the problem, emphasising that the

values of a, b and n can all be changed.


Figure 10 – The Task – Part 1

The second page introduces students to the expand function, which takes an

algebraic expression and multiplies it out. Some discussion of how to do this by

hand will have taken place when the problem was first introduced to the

students. The students will have experience of multiplying a single term over a

bracket and multiplying two brackets together by hand, but will not have tried to

deal with anything more complex than this.

At this stage the students are encouraged to experiment with changing the

values (primarily of n) using the calculator to carry out the expansion. The aim

here is for the students to get a feel for the shape of the problem, and to build

some idea of what the calculator is doing. This task could be seen as trying to

decode what the calculator is doing. With each example, the students are

giving the calculator an input and it is giving them a corresponding output; the

task the students are trying to solve at some level is to work out what it is the

calculator is doing in-between the input and the output.


Figure 11 - The Task - Part 2

When the students have had a chance to play with the calculator and get used

to entering expressions, they will be encouraged to move on and try some more

complex examples varying a, b and n. Examples are provided here to suggest

directions for the students to consider, but also to ensure that students are

aware of the full complexity of the problem they are being asked to consider.



Hopefully, students now have some idea of the problem, but they might also be

unsure about how to get closer to a solution. At this stage, the students will be

encouraged to ‘specialise’ and consider the simpler problem of the expansion of

and look at various values of n.


This page encourages students to look at this simpler problem systematically by

generating a series of examples that might help them see what is going on.



Some students might, at this stage, recognise the pattern and therefore not

need the next step. However, if they are still unsure, they will be encouraged to

consider just the coefficients of the polynomial. This step introduces the

function polyCoeffs, which returns the polynomial as an ordered list, from the

highest co-efficient to the lowest, including any missing values with a coefficient

of zero. The advantage of this approach for the student is that it naturally

presents this information in a manner that looks more like Pascal’s triangle.



Again, if students are struggling, the next page gives them a hint to lay out the

results in a triangle, and to generate the next few rows of the problem.


Next, the students are given the formal name for the result they have

discovered, and given hints about some of the other interesting properties of


this object. This leaves lots of scope for further investigation by interested

students. Students are asked to write a sentence to explain how they generate

Pascal’s triangle.


By this stage the students should have an hypothesis as to what the coefficients

of the expansion of would be for larger values of n. They will now be

asked to write down a prediction for this expansion when n = 7, and then use

their calculator to check their prediction.



Now that students have a working model of the coefficients of the expansion,

they can begin to rebuild the problem to include the added complexity that they

ignored when solving the simplified model. First they will need to reintroduce

the powers of x, which were discarded when considering only the coefficients.

Again the students will be encouraged to look at the results they had previously

obtained from the calculator, to make a prediction for the expansion of ,

and then test it with their calculator.



Again, building the problem back up, I return to the problem of expanding

, and remind the students that the original problem was broader than the

one they have currently solved.



The students will next attempt a systematic expansion of for small

values of n. With the results they have already found for the simplified problem,

students should, with some work, be able to extend those ideas to apply to the

broader problem.


Again, they are encouraged to make a prediction and then use the calculator to

test their prediction. At each stage, if their model doesn’t match the CAS

output, they will be encouraged to take another look at the results they have,

and see if they can find another possible model that might apply; make a new

prediction for a different value and then test it again.



In an attempt to see how well the students cope with writing on the keyboard of

the TI-Npsire, I have asked the students to explain the process they have

applied above. To do this I have used a ‘Q&A’ template. Explaining this

process is not a trivial task so it will be interesting to see how different students

explain it, and this is something I will return to in the analysis section.



The final section takes the ideas the students have already generated and

expands them to look at the expansion of . The problem is traditionally

expressed as just , with a becoming some multiple of the variable (in our

examples, x) and b becoming a constant, and so I have left it in that form. Here

the students are encouraged to explore the effect that introducing coefficients to

a and b has upon the problem, and again the hints in the example encourage

them to begin by looking at a simplified problem , which they can

compare to the results they found for , hopefully helping them see the

effect that the coefficient is having.


Next, they are encouraged to look at the expansion of to see what

effect changing the value of b has upon the problem. Then a few more

examples are presented, increasing the twos to threes and then exploring the

effect of changing both a and b at the same time.



Once they feel they have some idea of the solution to this problem they are

again asked to make a prediction and then check that prediction with the

calculator.



The final stage leaves the calculator behind and asks students in pairs to

produce a summary of their findings in the form of a poster. They will be

encouraged to refer heavily to their use of the calculator in this write up and

explain how they used the calculator to aid them.


The aim is that, by the end of this task, students should understand the process

that the calculator carried out for them in doing the expansion, and should be

able to carry out the expansion by hand without the aid of the calculator. In

essence the problem of expanding the binomial expression has been

transformed from that of an algebraic problem to one of a pattern spotting

exercise, within which the CAS provides the role of generating the examples

and verifying the predictions of the student.

So, in this example, it is possible to consider the CAS calculator as a Black Box

in the same sense that Heugl (1997 p.34) presented. The students investigate


what the Black Box is doing, and attempt to construct a model so that they are

able to carry out the work without the black box.

There is a possibility that students will replace the Black Box of the CAS

environment with the Black Box of Pascal’s triangle, without really gaining any

understanding of the connection between the two. It was my hope that students

would see the connections as they explored the problem. I consider the extent

to which the students were able to do this as part of my review to assess how

successful the task was in providing a conceptual understanding as well as a

functional understanding.

This task presents a way in which CAS could be introduced into the classroom

enabling students to approach problems from a perspective that would not be

available to them before the use of CAS.

Chapter 4 - Review the first implementation of the Activity and Refine the Activity 58

Chapter 4 - Review the first implementation of the

Activity and Refine the Activity

During the first lesson the students were each given a TI Npsire CAS calculator

and introduced to a small subset of the features available within the calculator.

The hope here was to help the students become accustomed to the calculator,

and also give them a flavour of its ability.

This lesson focused on the use of two functions - solve and expand – and

used the usual class textbook, “Formula One Maths book C3” published by

Hodder & Stoughton. Students looked at some exercises that they had

completed earlier in the year using traditional paper and pencil methods. The

students began by looking at some questions on solving linear equations using

the solve function. The purpose was to accustom students to the syntax

required by the calculator.

They started by solving simple linear equations, like , using the

CAS command Solve(4x+2=14-2x,x). This function has three main parts:

• Solve - This is the function name which tells the CAS system what it is

you want the calculator to, in this case solve something

• 4x+2=14-2x - This is the equation that you want the CAS system to solve

• x - Finally you have to tell the CAS system what variable you want the

equation solved for

Even with this relatively simple function there are numerous complications and

potential areas for confusion. One of the initial areas of confusion for the


students was where the Solve part had come from. One of the useful features

of most modern CAS environments is that you can choose between entering

functions using the menu structure or by simply typing the word.

The second confusion the students experienced was locating the ‘=’ sign. I

suspect that this was due in part to the fact that most ‘simple’ calculators still

use the ‘=’ button to mean ‘carry out this calculation’, rather than as the ‘enter’

key, which is used by the TI-Npsire CAS and most CAS systems. On the TI-

Npsire, the ‘=’ button is located on one of the small grey circular buttons just

below the ‘ctrl’ key.


Figure 28 - TI-Nspire CAS Calculator

Finally some students were unsure why they were placing an extra x at the end.

I suspect this may be due to their inexperience with more complex equations,

as most of the equations they will have met up to this point in their maths

education will have had only one variable and the solution will have been a

number. In these cases, the variable you are solving for is implicit and so is

never consciously stated. However, as the CAS solve function is more

powerful, it requires very careful use of its syntax.


With a little assistance from me as their classroom teacher, students were all

able to solve several equations using the calculator. This gave them some

confidence in using the calculator, and some idea of what it could do.

One interesting and unexpected result occurred when the students tried to

solve the equation . Upon entering the following command into

the CAS calculator - Solve(3(x-1)=3x-3,x) - the calculator returned the answer

TRUE. This caused some confusion amongst the students. I heard one

student say “It says TRUE, I must have entered it wrong” and then proceed to

try and enter it again. Some other students briefly looked at the problem

numerically and observed that they obtained the same value for both the left-

hand side and right-hand side, whatever value of x they chose. This, with a

little guidance, led them to the understanding that TRUE here meant that every

value of x was a solution of the equation as both sides were equivalent. There

is, I believe, potential for some very interesting mathematical investigation here,

using a combination of the Solve, Expand and Factor functions to explore the

concept of equivalent algebraic expressions. I feel that the CAS environment

could provide a very rich environment within which students could gain an

appreciation of equivalent statements. If this was coupled with the graphical

and tabular features of the CAS environment, then it would be possible to

explore this equivalence for multiple perspectives, and hopefully help students

acquire a more thorough appreciation of mathematical equivalence.

Having spent some time exploring the Solve function the students went on to

look at the Expand function, which would form one of the central features of


later investigational work into the Binomial Expansion. The Expand function

takes an expression and expands any brackets to give the expression in its

expanded form. For example if you enter Expand(3(x-1)) into the calculator it

will return the answer 3x-3. Again, the students looked at questions they had

previously solved manually as a way of helping them become accustomed to

the calculators. Hopefully, this also provided them with reassurance that the

calculator was doing the right thing, as they were able to solve the problems

manually.

As was to be expected many of the students tried expanding and solving

expressions which had no obvious meaning such as

Solve(abc+2+aa=bab+3,x). Which returned the solution as 0=-aa-abc+bab+1.

Other students tried expanding things like Expand((2x+1)^10293829) which

caused the calculator to pause for several minutes before they finally asked for

help and I had to remove the batteries to restart the calculator. This reaction

from the students fits with my observations when teaching ICT. Students tend

to like to try large numbers and unusual expressions in what seems like an

attempt to push the limits of the software/hardware. I have observed similar

things when using Dynamic Geometry packages with students for the first time

and observing that some students draw hundreds of points, lines or circles,

seemingly at random.

During the next lesson I introduced the structure and a basic outline of the

investigation. What was interesting in this lesson was the degree of excitement

and enthusiasm for using the calculators again. The degree of enthusiasm was

so strong at this point that it overshadowed the introduction to the investigation


as the students were so keen to start using the calculators and to fix various

problems that they found it hard to focus. This is a theme that I will return to, as

it is a recurring issue with using any technology in the classroom. In general,

there is still a marked hesitancy to allow unfettered access to technology in the

mathematics classroom when it is used solely to provide a novel experience for

the students. There are good reasons for this hesitancy, as one possible bi-

product of this novelty is that the emphasis for the student tends to shift from

the mathematics that the teacher thinks the technology is facilitating the

understanding of, to the technology itself. Students express excitement about

using Dynamic Geometry software, Graph Plotters, Graphical Calculators or

CAS systems because they find the novelty of the experience enjoyable.

However, because their use tends to be infrequent, students rarely become

conversant enough with the technology to allow its use to become transparent,

and so the use of technology often ends up being a distraction.

A similar idea about the issues of novelty in mathematics was presented by

Watson (2004) in her paper on dance and mathematics, in which she explored

the concept of novelty in maths education:

“However, such activities can also be a distraction from mathematics if they are

not integrated into the learners’ overall mathematical experience. At worst,

students only remember the [novel activity] …. In order for such experiences to

have more than novelty and motivational value, they need to relate closely to

classrooms in terms of what it means to do mathematics. Students need to be

able to pick up and use some of the ideas presented in ‘novel’ contexts in their

normal lessons, as habit. Innovation, at its best, makes a difference to how


learners think in all their experiences of mathematics, not just in the innovative

mode” (Watson 2004).

Whilst Watson here is actually discussing the use of kinaesthetic activities in

teaching mathematics, she could equally well have been describing the use of

technology. If the use of technology is not fully embedded into the students’

classroom experience there is a real danger that all they will remember is ‘going

to the computer lab’ or ‘using a graphical calculator’, rather than the

mathematical concept the teacher was hoping they would learn. This problem

presents a very real danger to the standard approach demonstrated by many

mathematics teachers of using technology irregularly and only allowing their

students to likewise use it when they see fit. This can result in both the inability

of students to use the technology independently and the clouding of students’

experiences with technology so that the students struggle to get past the entry

threshold and actually see the mathematics being explored.

My investigation here suffers from this issue because it was not possible to

obtain the calculators for a long enough stretch of time for the students to

become accustomed to them and, as the calculators were on loan, it was not

possible to allow students to use them outside of the classroom. These two

features made it much harder for the students to see the mathematics that they

are exploring. In an ideal world I would have liked the students to have had and

used the calculators for the majority of the academic year and to have been

able to take them home, but, in this study, this was not possible.

Returning now to the investigation of the Binomial Expansion, the students were

again each given a CAS calculator and this time were shown how to load the


pre-prepared document that was outlined in the previous chapter. Four or five

minutes were spent outlining the concept of binomial expansion, demonstrating

the process by expanding by multiply out two brackets at a time.

E.g.

I also indicated that to do this for larger powers would involve a large number of

steps and potentially several sheets of paper! Hoping to make the lesson as

investigative as possible I avoided giving too much guidance and focused on

answering technical issues or problems. To a degree, I think this worked quite

well, as, by the second and third lesson on this investigational work, the

students were starting to work more independently. However, the first lesson

lacked focus and many of the students, unused to this style of work, were

unsure what they were supposed to do. Many had worked through half the

pages in the calculator document within the first 10 minutes. But when asked

what they had found out they had no idea.

The second lesson was more directed and I encouraged students to focus on

looking at the coefficients using the PolyCoeffs function, and to generate and

write down examples and look to see if they could find a rule to generate these

numbers. Some students managed to do this in 10 minutes, but others took

several lessons and gentle guidance to get them to the solution. For me, one of

the most interesting things about approaching Pascal’s triangle (the pattern

generated by the coefficients of the expansion of ) as a pattern spotting

exercise, without any preconceptions as to how the numbers were generated,

was seeing the patterns that students did notice.


Most students quickly made observations such as, “There is always one more

number than the value of n”, and, “It is always 1’s down the first and last

column.” However, more complex observations also occurred, such as that “the

second column was just the whole numbers”, and that the third column was

“made by adding 1+2+3…” even though they couldn’t remember the name for

the triangular numbers.

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

Figure 29 - Triangle Numbers

Many of the students quickly noticed the symmetry of the numbers and could

therefore generate several rows of the pattern using a combination of

techniques without actually being able to describe the whole process. After

much exploration of these numbers and using the calculator to generate several

rows of results with which to work, students started to engage in the process of

creating and testing hypotheses. Students were encouraged to write down the

next row of the sequence using the pattern they felt they had found. Many

students however were not comfortable using the calculator to check their

hypothesis and asked me, as their teacher, whether their answer was correct. I

insisted that students go back to the calculator and use it to test their

hypothesis, by getting it to generate the next row of the pattern and comparing

this to the result they had predicted. Many times the calculator confirmed their

prediction but at other times it gave a different result to what the student had


anticipated and forced them to go back and revisit their ideas. This feedback

mechanism, whereby students could quickly check whether their ideas were

correct, led some of them to become more adventurous in the work. Some

students seemed happy to try several ideas and test them until they found one

that worked. However, some students still seemed hesitant to put anything

down on paper, in terms of predictions, until they were certain they were

correct. This more adventurous spirit, whilst not fully visible in this brief

experience with the work, does seem to point towards something similar to the

ideas we saw earlier expressed by Falbel (1992), where he considered the

effect on writing of using a word processor. My hope is that, given sufficient

time and availability, the CAS environment might also enable mathematics to

take on the “fluid, plastic substance that can be edited and manipulated at the

touch of a few buttons” and “liberate people to be more expressive and free” in

their mathematics. (Falbel 1992, p33)

Sadly, due to the time constraints of this particular project, it will not be possible

to study whether or not the CAS environment truly can achieve this increased

freedom and flexibility in the way in which students approach mathematics, but I

feel that there may be much to learn from further study into this area.

After a while, an understanding of a general method for calculating the next row

based upon the previous row (by adding the two numbers above) was reached

by all the students in the class. This was achieved in a variety of ways, as is

often the case in classroom-based investigation. About half of the group

reached this understanding either by themselves or working collaboratively with

students seated near them, while others needed more hints and direction to find


the solution themselves, and finally there was a small group of students who

waited until someone else had worked it out and asked them how they did it.

This last group of students were the ones who seemed to adapt least well to

this investigative environment; interestingly, a few of them were amongst those

who were normally most engaged and enthusiastic during lessons but, without

the obvious structure and traditional feedback system of the teacher

commenting upon and assessing their work, they seemed lost and unfocused,

unsure how to achieve ‘success’. How much of this is down to the preferred

style of the students and how much is down to experience is something that

perhaps deserves further study.

Returning to the investigational work, the next stage the students undertook

was trying to relate what they had discovered about the coefficients of the

polynomial expression to the expanded expression itself. This part of the

investigation turned out to be much harder than I had anticipated for a number

of reasons. This was partly because most of the students had not spent

sufficient time on the early stages of the investigation. As a result, they had not

appreciated the problem they were solving; they were seeing the numbers as a

numerical pattern and struggled to see how this was related to the expansion of

the binomial expression:

Figure 30 – (1+x)n

Binomial Expansions


A number of students were trying to use the same rule (adding the two terms

above) and so were asking me questions like “what is ?”. Here, and in

other places in this investigation, the lack of basic algebraic skills made it more

difficult for the students to engage in the activity at the level I had hoped. As a

result, many of the students ended up developing some techniques for

expanding binomial expansions of the form , but were not really sure

what these results meant. This lack of understanding of the basic concepts of

working with polynomial expressions, particularly on how to add, subtract,

multiply and divide simple algebraic expressions, meant that many students got

stuck working on ideas which led nowhere. One student commented that “he

knew what to do with the numbers but couldn’t work out how to add the x’s”.

However, one student described the method for expanding the expression as

follows: “you take the numbers from the triangle and then have a count-down on

the x’s”, by which he was trying to explain that the exponent of the x’s

decreased as you moved through the terms of the expression. He had split the

problem up and was seeing it as the result of the interaction of two separate

patterns.

Using techniques like this, most students were able to make predictions as to

the expansion of things like . One student continued to expand the

expression , and was delighted when he checked the result against the

calculator and saw that he was correct. That sense of joy, of having worked at

a problem and successfully found a solution to that problem, propelled him on

with renewed enthusiasm to approach the next stage of the investigation. This

sense of wonder and pleasure at something the student has created, and the


motivational property of this experience, which in this case is a model to

describe the expansion of , is central to the philosophy of

constructionism.

Once some of the students were happy describing the expansion of ,

they turned their attention to the expansion of . By this point many

students were getting a feel for how to approach this problem. Most students

began by generating examples which they could explore using the calculator in

the same way as they had for the previous two stages. However, again, some

students were clinging to the rule they had discovered in the first part of the

problem (generating Pascal’s Triangle) and so were trying to add together terms

from the row above:

Figure 31 – (a+b)n Binomial Expansions

One student asked me “How do you add 2ab to ?” and seemed confused

when I said that you couldn’t add them up in the way he wanted. I tried to

encourage this student to look for similarities between what had been found in

Pascal’s triangle and what he had found with the expansion of . I

suggested to another student, who was struggling to see what was happening,

that it might help if he wrote out his results side by side so that he could see on


the same line the coefficients from Pascal’s triangle, the expansion of

and , then to look for similarities and patterns between the results.

For some students this was enough for them to see clearly what was going on.

Again, I encouraged them to form a prediction, and then use the calculator to

test that prediction. Sometimes their prediction matched with the result for the

calculator, but, more often for this stage than with the previous stages, students

had either made error in the calculation, or had not really correctly interpreted

what was happening with indices for a and b.

Language was an issue for many students who felt, although they could write

out the next line of the expansion, they did not feel they could describe to

someone else what they had done. Phrases, such as “the a’s count down

whilst the b’s count up”, were used as students tried to explain to each other

what they had found.

A small number of students progressed to looking at more complex examples of

expanding binomial expansions, and most of these attempted an exploration of

the expansion of . By this stage, the students had established for

themselves a methodology within which to approach this question and began by

generating the expansion of , for n=1, 2, 3, 4 & 5, and comparing this to

their earlier results. At this point in the investigation I felt that their lack of

algebraic experience hampered their progress more than at other stages. With

the expansion written out, one student isolated the coefficients (recognising as

he said that the x’s were still just decreasing). This gave him the following

pattern:


2 1

4 4 1

8 12 6 1

16 32 24 8 1

32 80 80 40 10 1

Figure 32 - (2x+1)n Coefficients

He could see some similarities between Pascal’s triangle and this new pattern,

but could not see how this result might have been related. His first step was to

reverse the result given to him on the calculator, so that the line of ones

matched up with the line of 1’s on Pascal’s triangle, and he laid it out in the

more usual trianglular pattern. He then observed that the second column was

twice the size of the equivalent column on Pascal’s triangle. But, rather than

continuing with this line of thought to compare the lines of this new pattern with

those of Pascal’s triangle, he returned to looking for patterns which were similar

to those used to derive Pascal’s triangle. He now experimented with

combinations like multiples of one number added to another number. After

trying several combinations he tried the following rule “Twice the number the

right (above) added to the number on the left (again above)”. This rule enables

you to generate a Pascal related triangle for . Using the non-standard

CAS approach enabled this student to find a solution to the problem that he

might otherwise have missed.

Due to the lack of time at the end of this project, the student in question didn’t

have enough time to explore why this result might work or to explain how this

might be extended for different values of the coefficients of x or the constant.


An indication of why this method works can be seen by rewriting the coefficients

of the expansion as powers of 2. This gives us the following:

2 1

1

1

1

Figure 33 - (2x+1)n Coefficients in terms of Pascal's Triangle

So, by multiplying the number on the right by 2, both numbers are now multiples

of the same power of 2 as the number beneath. This means that you can factor

this out and simply add the coefficients which are just the normal ones from

Pascal’s triangle, hence giving us the desired result.

I would have liked to see where he could have taken this observation and I feel

that, given more time, he might have been able to fully describe this relationship

and possibly extend it to deal with the full expansion for any value of coefficient

of x or constant within .

One potential issue with working with CAS systems, but CAS calculators in

particular, is that there is not the same need for the student to write things down

as they carry out their work. I encouraged the students to make rough notes in

their books to remind them what they had looked at in the previous lesson and

to use a planning document for the next stage. I used a similar approach here

to that advocated by Harel & Papert (1991 p44). In their Instructional Software

Design Project, Harel and Papert gave the students ‘Designer’s Notebooks’: in


these students “wrote about the problems and changes of the day… and

sometimes added designs for the next day” (p44).

The purpose of these notes, for the students and myself as the researcher, was

to tie together the project as it was being carried out over a couple of weeks in

short (35 minute) lessons. It was interesting to see what students deemed

significant enough to include in their notes. Many students chose not to write

very much and, instead, mainly included lists of results and vague sketches and

comments, which were sufficient to remind them what they were doing.

Below are some examples of the notes the students made as they went through

their work. Ball (2003, 2004) and Ball & Stacey (2003, 2005) have studied

extensively what students write when working in a CAS environment. Particular

emphasis was given to examining whether CAS notation should be permissible.

An example that they consider is finding the missing side of a right-angled

triangle using Pythagoras’ Theorem. Here the problem can be solved using

the following CAS notation: Solve(c2=a2+b2 | a=3 and b=5, c). However, with

this, there are no intermediate steps. Ball and Stacey then raise the question:

“Should the teacher accept this written record which (a) shows no intermediate

steps and (b) uses the calculator syntax directly?” (Ball & Stacey 2005 p117)

There is still much work to be done in this area to appreciate fully the effect that

CAS may have upon how students communicate mathematically. However,

one observation made by Ball & Stacey, based upon their research, is that

students write solutions using CAS


“that will contain more words and use function notation more. Some

changes may occur in the mathematical notation that is regarded as

acceptable, but there need be no fears that students will replace

standard notation by incomprehensible machine-speak.” (Ball & Stacey

2003 p93)

Returning to the students in my study, I now examine some of the students’

notes produced during this investigation.

Student A:

Figure 34 - Student A's notes - Part 1



These notes from Student A show that he was initially excited about using the

calculators, describing them as “mini-computers” and “very understandable”.

However, as he progressed through the investigation, he seems to lose

confidence in the calculator, using the phrase “still not alright with them”, and

finally, by the end, describes the CAS calculator as “confusing”. Despite this

lack of confidence, he manages to demonstrate the effective use of the

calculator, as can been seen from his other notes. I am hopeful that, given

more time and experience with the CAS system, he could regain his confidence

in its use.


Student B:

Figure 38 - Student B's notes - Part 1



Student B has been much more explicit in his working and has documented his

steps throughout the process. He makes an interesting association between

how he feels about the calculator and how he feels about the maths he is

investigating: “I found the calculator pretty easier and the more I understand the

work the better it is to understand the calculator”.


This raises the question of whether Student A really found the CAS calculator

confusing, or was it the maths he found confusing, which effected his

perception of the calculator.

Student C:

Figure 41 - Student C's notes


Student C has followed a similar approach to the other students. The students

were not told what to write down, and were told that they should write down as

many notes as they found useful (although they were encouraged to write a

summary of how they had found the session at the end of each lesson). It is

noticeable that all the students chose to write something down as they went

through the investigation. I would be interested to carry out this investigation

again, but using a computer-based CAS environment, to see if one of the

reasons for writing things down as they went was the size of the screen and

therefore the amount of information the students could see at one time.

Student D:

Figure 42 - Student D's notes - Part 1


Figure 43 - Student D's notes - Part 2


Student D showed a progression throughout his work, from a calculator-based

notation, such as ((1+x)1) = {1, 1} at the start of his notes. Later, he realised

that the curly brackets denoting the set were unnecessary, and so he stopped

including them. By the time he was looking at the expansion of (a+b)n, he was

no longer writing down the superfluous brackets from the CAS commands

PolyCoeffs and Expand.

Student E:

Figure 44 - Student E's notes

Student E articulated a feeling experienced by many when they first used a

CAS calculator. Whilst there was often a degree of excitement, there were also

reservations, because the machine looked very complex; hence the experience

that the calculator was “confusing at first”. However, because our task was

limited to a small subset of CAS functions, the student quickly “got the hang of

it”.


After completing the practical work, I wanted the students to summarise their

findings in the form of a poster explaining their findings. The students were

given an open brief of summarising whatever they had found out about either

binomial expansion or the CAS calculator, though most decided to cover both

elements at least in part.

Figure 45 - Poster 1


In this first poster, the students chose to document the process they undertook,

but they did not give very much insight into their findings. Notice in this poster

the use of the CAS vocabulary in points 1 and 2. However, the later points,

whilst explaining the same ideas, do not use the vocabulary.


This second poster again draws attention to the calculator notation. This time,

however, the students give us some clue as to the link between the expansion

and Pascal’s triangle.



This poster does not show any sign of calculator notation and, if you removed

the comments about CAS in the top right corner, it could easily be mistaken for

a project completed without the aid of CAS. This group of students have tried


hard to explain the patterns they spotted and the relationships observed

between Pascal’s triangle and the Binomial Expansion.


This poster demonstrates much more clearly the observations made by one

group. In this artistic poster, they explain the generation of Pascal’s triangle,

and go on to describe the powers of x in as being a “countdown”.

Whilst they lack some of the vocabulary to fully describe their observation, they

have, with the aid of CAS, been able to generate suitable examples and to

observe and describe the results.

There may be some concern that the students have, in the most part, simply

replaced the ‘black box’ of the CAS calculator for another ‘black box’ of an

algorithm to expand binomial expressions. However, my hope is that, when the


students return to this topic in several years’ time, they will better recall the

patterns and relationships having discovered them for themselves, and will be

better equipped to understand why they are related because of their

understanding of the patterns.

Chapter 5 - Explore results and look to see what general statements 91 we can say about what might work in the future

Chapter 5 – Discussion of Results

The purpose of this experiment was to explore one possible method for using

CAS within Secondary School education. Through it I hoped to demonstrate

both that secondary school age students were capable of using CAS effectively

and also that, through the use of CAS, the students would be able to access

mathematics that would previously have been inaccessible to them due to the

algebraic demands.

Looking at the experiment described above, I am confident that I have shown

that secondary school pupils are capable of effective use of CAS systems and

that, through this use, the students were able to carry out a piece of

mathematics usually not studied until 4 years later, and then only by students

studying A-level mathematics.

There were many positive outcomes from this experiment. The students

generally engaged with the technology in a focused and meaningful way.

However, this increased motivation and focus (in comparison to their behaviour

in their normal mathematics lessons) may be attributed to the Hawthorne Effect

rather than to the use of the CAS system. A much longer trial would be

necessary to determine if the increased motivation was due to the novelty of

using the CAS calculators, and the fact that the students knew they were taking

part in a study, or was actually due the CAS calculators themselves.


One observation worth exploring was the differing lengths of time it took for

students to feel comfortable using the technology, and this is something I return

to in the final chapter. While some students quickly adapted to the specialist

syntax associated with the use of CAS, many others struggled initially with even

the simplest of commands, such as expand(), failing to balance the number

brackets correctly and entering commands like expand(1+x)^5 which returns

rather than correctly entering expand((1+x)^5). However, by the end of

the short series of lessons, all students were confidently using the expand(),

and PolyCoeffs() functions. In conclusion, the introduction of CAS into the

classroom, like most technology, has a trade off between the time saved by

using technology and the time lost through training the students to use the

technology.

In order for the technology to become fully integrated into the students’

repertoire of skills, a significant amount of time and investment is required on

the part of the teacher and the student. Any move towards increased use of

technology in the classroom must be coupled with consideration of the time

required to become proficient in its use. From my own teaching experience the

problem can easily become a vicious circle, where teachers don’t use a

particular piece of technology (e.g. Dynamic Geometry) with students because

of the amount of time it would need for students to become competent users;

because the students get no experience, they remain novice users who require

additional time and support when the technology is used. The net result is that

it is often quicker (in the short-term) to achieve the desired teaching point

without technology than it is to invest the time to train the students correctly in

its use then apply it to the situation in question. Consequently, students


continue not to get the regular experience they require and every time they do

use that particular piece of technology they have to be retrained.

Another observation worth looking at was varying levels of engagement that

students demonstrated. Some students thrived from the freedom CAS gave

them to explore a problem without constant external guidance from the teacher.

These students continued to interact with the teacher but not in the way they

would normally do in a maths lesson. Instead of coming up to ask “how do I do

this?” or “what should I do next?” they were coming up to share their success,

saying things like “I got it!”. For these students, the teacher became a facilitator

in the work, sharing in the successes and suggesting avenues for further

exploration, rather than simply an expounder of information. These students

were able effectively to change roles within the task, from being the student

exploring the task, to providing technical support to other students, becoming

peer-teachers, explaining both the ‘how’ and ‘why’ questions that other students

asked. For these students, the CAS environment, and the more open

exploratory style of activity that that they were pursuing, enabled them to push

beyond the constraints of the teacher’s expectations.

However, it must be acknowledged that, for some pupils, the transition was

much less smooth. A significant number of pupils found the lack of structure

associated with the open task disconcerting and were unsure exactly what was

expected of them. This group of students spent a considerable amount of time

asking for clarification and direction, and were keen to check that their progress

was correct. For the students who formed my study group this is likely to be

partly due to the traditional approach usually experienced in maths lessons,


where the teacher acts as the gate-keeper to learning and the judge of what is

correct. In this more open task, where the students were asked to judge some

of these ideas for themselves, using the CAS environment as the authority with

which to check their predictions, this group of students felt confused and

abandoned. There were even some suggestions that it was the teacher’s job to

“tell them the answers” and that I was in some sense not doing my job properly.

This phenomenon is not unique to the CAS environment and is really an issue

related to the more open style of task that I believe is likely to accompany a

move to a greater effective use of CAS. Whilst I am sure it would be possible to

use CAS in the traditional instructional approach, the most effective use, I

believe, is likely to belong in the more open ended use of the technology,

following the constructionist approach (Papert 1980, 1991, 1996) discussed in

Chapter 2 and demonstrated through the task carried out by the students. If

students are only required to answer the type of questions we currently ask but

are allowed to use CAS, not only are they likely to lose many of their algebraic

skills, but they are also unlikely to develop any additional skills. Thus, the use

of CAS in a more traditional instructionalist environment is likely to lead to the

loss of algebraic skills that is feared by some (Buchberger 2002 p.5). Coupled

with that, CAS tagged on to a traditional teacher style and curriculum is likely to

be very uninspiring for both the teacher and the student. This discussion leads

into the final chapter in which I consider the possible shapes of a CAS-enabled

curriculum.

Chapter 6 – What belongs in a CAS Curriculum and how should we assess it? 95

Chapter 6 – What belongs in a CAS Curriculum

and how should we assess it?

In this final chapter I will explore the issues affecting assessment and its

influence upon a CAS curriculum. Building upon what we have seen so far

about the use of CAS in chapter 2 and the example of CAS tasks I explored in

chapters 3, 4 & 5 I will now look at how these ideas might be used within a CAS

curriculum.

“Look around you in the tree of Mathematics today, and you will see

some new kids playing around in the branches. They’re exploring

parts of the tree that have not seen this kind of action in centuries,

and they didn’t even climb the trunk to get there. You know how they

got there? They cheated: they used a ladder. They climbed directly

into the branches using a prosthetic extension of their brains known

in the Ed Biz as technology. They got up there with graphing

calculators. You can argue all you want about whether they deserve

to be there, and about whether or not they might fall, but that won’t

change the fact that they are there, straddled alongside the best

trunk-climbers in the tree – and most of them are glad to be there.

Now I ask you: Is that beautiful, or is that bad?” (Kennedy 1995)

Back in 1995 Dan Kennedy wrote a wonderful allegory about the way in which

Graphical Calculators would allow students to bypass some of the traditional


requirements of mathematics; this is even more true of CAS systems. The

question still remains though, is this desirable?

This question is the crux of the debate about CAS use. But to answer this

question we have to delve even further into a discussion about the very nature

of mathematics, for it is at this level that fears about CAS lie. Some fear that a

greater use of technology within mathematics will devalue the algebraic skills

that they see as the very essence of mathematics (Taylor 1995 p.82, Gardiner

2001 p6-8). For others mathematics is about logical proof and, for them, the

use of a computer in general or a CAS system in particular causes great

concern (Burchberger 2002 p.5) as can be seen from the fall out after the

computer-assisted proof of the Four Colour Theorem (Detlefsen & Luker 1980).

However, in recent years there is a growing appreciation that the computer has

a place within mathematics. Additionally, there is a third group who believe that

the central tenant of mathematics (or at least school mathematics) is the

concept of problem solving (NCTM 1989 p.6, Lubienski 2000); for this group,

CAS is seen here as another tool, alongside traditional algebraic and geometric

tools, which can be brought to bear upon a mathematical problem (Kutzler 1999

pp.9-10).

At this stage, it is worth looking again at the potential benefits and pitfalls of the

use of CAS in secondary mathematics education (Leigh-Lancaster 2008 p17).

I will begin by looking at the potential benefits, which include the opportunity to

revisit the order and content of the current mathematics curriculum. CAS could

potentially enable curriculum content to be ordered by conceptual difficulty,

rather than the current ordering which is based more upon computational


difficulty (since the current syllabus is in the most part a legacy from the pre-

technology era). I have already demonstrated that this type of reordering is in

principle possible through the task I explored in chapters 3,4 and 5 where I took

a topic usually studied at A-level and carried it out with a class of year 8

students (4 years early). CAS may also enable the teaching of important

mathematical ideas that without CAS are too difficult to teach effectively; for

example, a more detailed study of differential equations might be possible.

CAS, as I hope I have shown in my small study, can also be used as a vehicle

for mathematical investigation. CAS offers the opportunity for students to

“transcend the limitations of the mind” by generating a much greater number

and range of examples (Heid 2003 p36-37). The concept of programming as a

means to teaching mathematics has been around almost as long as computers

have been in schools, but was popularised by Seymour Papert with LOGO.

CAS provides us with another programming environment which is designed for

mathematical use.

Another area of interest is the idea of working with multiple concurrent

representations of the same concept to emphasise the inter-relatedness of

mathematical representations. CAS has always enabled greater links between

topics but a recent CAS platform, the TI-Npsire CAS, pushes this idea even

further through a document model, which allows the same functions and

variables to be used in the dynamic geometry/graphics window, in the

table/spreadsheet view and the calculation view.

CAS would enable long and technical calculations to be completed by the CAS

system, thereby enabling the students to concentrate on more conceptual areas


of the topics being examined. CAS also provides an immediate feedback

mechanism for students independently to check and monitor their ideas.

Since CAS is not limited by the calculation difficulty that people experience

when dealing with calculations by hand, users would not remain limited to

simple examples of topics (which can be carried out by hand). Instead, users

could model problems and situations in more complex and realistic ways.

One final benefit is that CAS requires the student to use a precise syntax. This

means that students are forced to clarify their mathematical thinking so they can

express their ideas in a form that the computer can understand. The problem is

that students are also required to learn a new complex syntax, something that

will take time.

On the other side of this argument, there are obviously some pitfalls associated

with the use of CAS. Some of these, as already mentioned, are related to

ideology, but other potential problems include the degree to which students’

knowledge and skills with important and valued conventional, paper and pencil

and mental techniques may atrophy through lack of use. Which brings us back

to the question I looked at in Chapter 2 raised by Herget et al (2000) as to what

extent, if any, these traditional paper and pencil techniques will continue to have

a place in a CAS curriculum.

There is also the concern as to how students who struggle with the current

curriculum would cope with a more conceptually demanding CAS curriculum,

which may have less emphasis on the calculation skills and more on the


interpretation skills required to construct a mathematical problem and review its

solution. There is also a concern that a CAS curriculum may reduce the role of

the teacher “in terms of traditional (and valued) pedagogy” (Leigh-Lancaster

2008 p17).

So assuming, like me, you find that the arguments in favour of using CAS

outweigh the potential pitfalls, let us move on to look at the obstacles to its

implementation. CAS systems and calculators are already here. They are

available to those who want to use them but, currently within the English

mathematics education system, we are still for the most part pretending they

don’t exist. The approach in this country has been to ban the technology. It is

currently not allowed in any school exams, rather than grapple with the

consequences of it. A similar approach can be seen with the way in which A-

level exam papers are written in a broadly graphical calculator neutral manner,

so as to remove any advantage graphical calculators might provide. But this

also has the bi-product of making graphical calculators unnecessary, which is

one of the reasons they are not widely used at A-level. The following quote is

taken from the recent QCA report into participation in A-level Mathematics:

“Most centres do not provide or choose not to use graphic calculators

and most students do not have their own.” (Matthews & Pepper

2006 p64)

Contrary to this the International Baccalaureate requires the use of a Graphical

Display Calculator (GDC) by all its students. This has been a compulsory part

of three of the four diploma courses since 1998, and all four since 2004:


“From 2001 onwards, students using only four-function scientific

calculators or early versions of the GDC were at a disadvantage in

examinations. Examiners set questions assuming that all students

had a GDC with the minimum functionalities.” (IBO, 2005)

So, for the time being, change to the major assessments of GCSEs and A-

Levels in England is going to be dependant upon a number of factors, including

the willingness of exam boards to write questions for which using CAS is

beneficial. This has implications for accessibility as students cannot be

disadvantaged for not having access to equipment; central funding would be

required to ensure that all students have access to the technology. Another

major obstacle to the uptake of CAS is the teachers. Most mathematics

teachers have little or no experience of CAS, so its effective implementation will

involve a significant amount of professional development.

However, if its introduction is to be successful then the use of CAS must be

compulsory. Established teachers are likely to resist significantly changing their

approach, so, without the incentive of their students being required to use the

CAS system in the exam, uptake is likely to be similar to the current situation

with Graphical Calculators, where the technology is neither necessary for

success in the course nor, as a result, widely used.

In order to explore what a CAS curriculum and its assessment might look like, I

am going to assume that these practical details of equipping students and

training teachers can be overcome.


Before delving too deeply into the features of a potential CAS curriculum, I will

begin by looking in some more detail at what is already being done in other

countries (VCAA 2003).

The first country to implement a CAS examination structure was the US College

Board for the Advanced Placement Calculus exam. They have, since 1995,

taken a CAS-neutral approach, where a broad range of graphical and CAS

enabled calculators are allowed in the same exam. For this implementation,

CAS should not provide any advantage.

However, it is important to consider whether it is possible to create a paper

which is truly equitable to students who have access to CAS and those that do

not. As Cannon and Madison (2003 p310) observe, the “practice of allowing,

but not requiring, specific computer capability complicates assessment and

makes inequities very difficult to avoid”.

The second country to implement a CAS examination structure was Denmark

through their Baccalaureate examination. A pilot scheme ran between 1996

and 1999, and this has been generally available since 2000. There is a

technology-free component and an examination which assumes access to

graphical calculator or CAS. Some questions are common to both CAS and

graphical calculator papers and some distinctive to the technology used. This

technology-enabled exam is also an open-book exam. It is expected that, in the

near future, CAS will be assumed for all written mathematics in the

‘Gymnasium’; however, the assessment will retain a paper and pencil element.


Switzerland has a teacher-constructed examination system, which has allowed

for the use of CAS calculators or computer-based CAS systems at the teacher’s

discretion since 1998. These exams, similarly to those in Denmark, can also be

open-book or allow access to electronic files.

In 1999, both France and Austria allowed the use of CAS. In France, CAS-

neutral questions are used, while allowing unrestricted access to CAS through

all examinations. In contrast, Austria has a similar approach to Switzerland,

whereby teachers construct their own examinations, including both technology-

free and CAS-permitted components. Again, CAS is permitted at the teacher’s

discretion.

More recently, between 2001 and 2005, the state of Victoria in Australia ran a

three-stage pilot with assumed access to approved CAS for all parts of the

examination. I return to the VCAA (Victorian Curriculum Assessment Authority)

courses in more detail in the next section.

The International Baccalaureate ran a two year trial, between 2004 and 2006, of

the Higher Level Mathematics, which assumed access to an approved CAS

calculator. The exam included both CAS assumed and non-calculator

elements.

The development of a CAS curriculum in Victoria, Australia, has been very well

documented from the initial pilot studies through to its final rollout to all

students, which is due in 2010. To illustrate how CAS could be introduced in

England, and to gain some insight into what a CAS curriculum for England


might look like, I describe the details of the VCAA's implementation (Leigh-

Lancaster 2008 p21).

I begin by looking at the assumed technology for examination in Victoria from

1968. From 1968, it was assumed that students had access to slides rules and

four-figure tables. From 1978, scientific calculators were assumed. From 1997,

scientific calculators with bi-variate statistical capacity or approved graphical

calculators were permitted; the examinations were written to be graphical

calculator neutral. From 1998, there was assumed access for graphical

calculators in Mathematical Methods (similar to Mathematics A-level in England)

and Specialist Mathematics (similar to Further Mathematics A-level), and

graphical calculators were permitted for Further Mathematics (similar to the Use

of Mathematics A-level in England). From 2000, it was assumed that all

candidates had access to a graphical calculator in mathematics examinations,

and examinations incorporated a question that would require the use of

graphical calculator functionality.

From 2001, a pilot study of Mathematical Methods assumed access to

approved CAS technology in pilot examinations. Between 2006 and 2009,

there was assumed access for approved graphical calculators or CAS, as

applicable for the relevant course. From 2010, it will be assumed that

candidates have access to approved graphical calculators for the Further

Mathematics course and to approved CAS technology for Mathematical

Methods and Specialist Mathematics.


As can be seen, new technology has been a constant part of the VCAA

curriculum and assessment strategy. However, unlike the English A-level

system, technology usage has mostly been ‘assumed access’ rather than

‘permitted access’ (Leigh-Lancaster 2008 p21).

“The VCAA, and former Board of Studies’, strategic decisions to

assume student access to an improved graphics calculator for VCE

Mathematics examination (and hence as the enabling technology) in

1998; and to subsequently undertake a CAS based pilot study for a

Mathematical Methods subject, were based on cognisance of

general developments in society, the discipline, education and

technology, as well as being informed by more specific

developments in various educational jurisdictions and systems over

the last decade” (Leigh-Lancaster 2008 p21 emphasis original).

Now, before looking in detail at the course content and assessment structures, I

consider the pilot studies carried out by Mathematics Education Research

Group of Australia (MERGA) for the VCAA. Between January 2001 and

December 2005, the VCAA offered an accredited pilot study of Mathematical

Methods (CAS) units 1 - 4. By the end of the extended pilot in 2005, several

hundred students from a range of urban and rural, co-educational and single-

sex schools across the state, catholic and independent sectors were involved

using a range of approved hand-held and computer based CAS (Evans et al

2005 p330).


The Mathematical Methods and Mathematical Methods (CAS) units 1-4 involve

the study of the following major topic areas: Co-ordinate Geometry, Functions,

Algebra, Calculus and Probability. Units 1 & 2 are usually taken at the end of

Year 11 (Year 12 - Lower 6th England) with Units 3&4 at the end of Year 12

(Year 13 - Upper 6th England). The Mathematical Methods (CAS) modules

were developed from the current Mathematical Methods course. Therefore the

pilot Mathematical Methods (CAS) units were a “natural evolution of the

Mathematical Methods Units 1-4” (Leigh-Lancaster 2008 p22 emphasis

original).

The pilot examinations for Mathematical Methods (CAS) used the same

structure and format as the existing Mathematical Methods course and were

held concurrently. Examination 1 was designed to test facts, skills and

standard applications and took the form of an hour-and-a-half exam, which

contained multiple-choice and short-answer questions. Examination 2 was an

analysis task which again took the form of an hour-and-a-half extended-

response examination.

Central to the VCAA’s approach to the pilot was the development of resources

to accompany the course. Initially, the VCAA developed a sample paper for

each examination as well as a bank of related supplementary questions, with

solutions and comments, which were made available online. They also made

provision for targeted professional development to teachers from schools

involved in the extended pilot study.


In the analysis of the Multiple Choice section of the Pilot study it was seen that

on “15 of the 21 common multiple choice questions, a higher percentage of the

CAS cohort of students obtained the correct answer” (Evans et al 2005 p331).

In fact, for the questions which were classified as technology-independent, the

CAS cohort outperformed the non-CAS cohort on 9 of the 13 questions.

An important observation from this study was that, whilst CAS offers high levels

of reliability on some simple questions, it is not the case that all mathematical

tasks can be classified as ‘trivial’ because of the access to relevant CAS

functions. This can be seen by looking at the following question on the 2004

Mathematical Methods (CAS) Examination 1 paper, which asked the candidates

to factorise a cubic expression.

Figure 49 - Mathematical Methods (CAS) Exam 1 - 2004 - Question 9

However entering the CAS command factor(ax^3-bx) will return x, ax^2-b. For

students to correctly answer the question still requires a degree of interpretation

of the question, such as a selecting the relevant field over which factorization is

to take place and the ability to use CAS effectively to solve the problem. This

explains, in part, why only 61% of the CAS-cohort correctly answered this

question compared to the 59% of the non-CAS group (Evans et al 2005 p332).


As David Leigh-Lancaster (2003 p6) points out:

“Access to a given functionality or repertoire of functionalities, does not

necessarily confer their effective use in practice, or understanding of

important ideas and principles underpinning such use.”

For this question the student should have used the command factor(ax^3-bx,x)

to signify that they wanted the expression factorised over the linear domain,

which would have returned the correct answer of E.

Significant improvements were seen in the performance of the CAS group in

each of the three years of examination during the pilot study when solving

questions which asked students to find the intersection of two curves. This fact

may support the idea that the requirement for careful use of syntax of symbolic

expressions may encourage a similar attention to detail more broadly with other

functionality.

In the common questions from the short answer section of Examination 1, Leigh

Lancaster (2003 p8-9 & appendix) observed that CAS students out-performed

the non-CAS students in all three questions. In the CAS-only questions from

the 2002 Mathematical Methods (CAS) Examination 1, it was noticeable that,

whilst most students were able correctly to formulate a suitable definite integral

expression and related equation for question 1 (see figure 50 below), this did

not lead to corresponding correct calculations or evaluations to “anywhere near

the same extent” (Leigh-Lancaster 2003 p8).



The second Examination papers contained more extended answer analysis

questions. The CAS-cohort continued to perform equally well or outperform the

non-CAS students on all the common elements. During the 2002 pilot

examination (Leigh-Lancaster 2003 p9), where numerical computation or graph

sketching was required, both groups “performed comparably, or the CAS cohort

performed slightly better” (2003 p9). Where algebraic manipulation was

involved, access to CAS “improved the reliability of correct responses” (2003

p9). Where calculus and algebra were both involved in a theoretical context,

again access to CAS improved the reliability of correct responses.

On the question involving probability, which involved binomial, hypergeometric

and normal distributions with mainly numerical calculations, the CAS cohort

performed slightly better. This was despite the fact that the examiners

expected there would be “no appreciable difference between the two cohorts

here” (2003 p9).

On one question, from the 2004 Mathematical Methods (CAS) Examination 2

paper, the CAS students were asked to describe a sequence of transformations


which map a function onto the graph of ,

then find the x-axis intercepts in terms of k, and finally find the area of the

region bounded by the graph of and the x-axis in terms of k.

Students on the non-CAS exam were asked to answer the same question but

for the fixed value of k = 2, that is, to solve .

Evans et al (2005) comment that:

“two cohorts were similarly able to describe a suitable sequence of

transformations with similar levels of success, the CAS cohort were able

to tackle part ii [x-axis intercepts] and iii [area bounded by the x-axis]

more successfully notwithstanding the use of a parameter rather than a

specific value.” Evans et al (2005 p334)

On the CAS-only questions the introduction of additional variables, or

parameters, to problems was used. In one question, students were required to

find the largest value of a parameter k for a given, but unknown, value of T,

such that an equation has a solution over the domain of the underlying

modelling function. The average mark on this question was just 0.21 out of a

possible 2 marks, and matches the experience of the Danish pilot study. The

chief examiner of the Danish pilot remarked that “based on their experience,

such questions are challenging to students, and what appears to be a small

increase in complexity of question design can be a substantial increase in

difficulty for students, especially where a parameter is involved” (Leigh-

Lancaster 2003 p10).


An additional finding, which is worth noting from the VCAA pilot studies, came

from early in the study. With the initial three schools in the 2001 and 2002 pilot,

the VCAA undertook an algebra and calculus skills test related to the content

covered to that stage, with both the Mathematical Methods and the

Mathematical Methods (CAS) cohort. These skills tests were undertaken

without access to either a graphical or CAS calculator. The results clearly

showed that “students from the CAS cohort were able to perform as well or

better on this material as the non-CAS cohort” (Leigh-Lancaster 2003 p10).

In chapter 2 I looked at how CAS may trivialise some current A-level questions.

I also looked at some of the techniques used to remove the advantage of CAS,

creating CAS-neutral questions. So in this next section I will examine some

exam questions from CAS-enabled syllabuses to identify areas of commonality

between the questions and explore how they differ from non-CAS questions.

Use of Parameters:

The first area of commonality is one that has already been mentioned in

passing and that is the ‘use of parameters’. A good example of this comes from

the VCAA Mathematical Methods (CAS) - Examination 2 - 2006 paper Question

4:



In this question, students are repeatedly asked to express their answers in

terms of parameters. In some parts of the question, students are given


additional information to enable them to work out the value of some of these

parameters.

Another useful example of the use of parameters can be seen in the following

multiple choice question from the VCAA Mathematical Methods (CAS)

Examination 2 - 2007 paper, which takes a typical simultaneous question,

introduces a parameter, and then asks the student to state when the solution is

unique:


Another example of this can be seen on the Danish STX December 2007 exam

paper.

Figure 53 - Danish STX December 2007 - Question 11

This question asks students to explain the relationship between a and c when

there is only one solution.


The technique of introducing additional parameters was also used in the

American College Board AP Calculus exam papers. However, in this case the

question setters sought to use parameters to remove the advantage that CAS

might provide. An example of this approach can be seen in 2006 AP Calculus

BC Free Response Paper (Form B).

Figure 54 - AP Calculus BC Free Response Paper (Form B) 2006 - Question 3

How successful this technique is in removing the advantage of CAS is not clear,

although, in the author’s opinion, this approach is unlikely to level the playing

field between CAS and non-CAS users. This seems to be, at least in part, the

opinion of the paper setters as well, as they felt the need to include the line

“Show the work that leads to your answer”.

The technique of requiring candidates to show working is also used in the

English A-level system as a method of making questions graphical-calculator


neutral. The following question, taken from an MEI Core 3 Paper from June

2006, illustrates this.

Figure 55 - MEI Core 3 June 2006 - Question 6

In part (iii), the student is required to find the half-life in terms of the parameter

k. However, this style of question appears to be used more often in CAS-

enabled exams.

Step-wise Functions:

Another common feature of CAS examinations is the use of step-wise functions.

Step-wise functions introduce an additional level of complication, which results

in the ability to model more complex and realistic situations. An example can

be seen in the following question from VCAA Mathematical Methods (CAS) -

Examination 2 - 2007 paper.



Use of Restricted Domains:

Restricted domains are often used in conjunction with questions about when

functions are invertible, or meet some other criteria. An example of this style of


question can be seen in the following multiple choice question from VCAA

Mathematical Methods (CAS) 2005 - Examination 1 Paper.


Contextualised Questions:

A significant increase in the use of contextualised questions is common in CAS-

enabled exams. Examples have been shown above to demonstrate the other

common features.

The following question is a highly contextualised example, which formed part of

the 2001 final exam in St. Pölten, Austria (cited Böhm et al 2004 p101-102).

The question is about describing a road to be positioned near an 18-hole golf

course.


Figure 58 - St Pölten, Austria 2001 Final Exam


A further example of such contextualisation can be seen in the following

question taken from the VCAA Mathematical Methods (CAS) Examination 2

2005 paper.

Figure 59 - Mathematical Methods (CAS) Exam 2 -2005 - Question 3 - Part 1


Figure 60 - Mathematical Methods (CAS) Exam 2 - 2005 - Question 3 - Part 2

This question, about working with harmonic functions, is placed in a context of a

problem about the construction of a bridge. Unlike some ‘contextual’ questions,

this problem carefully refers to the context throughout and the questions are


phrased in terms of lengths, and costs. Students need to do more than just

carry out mathematical operations, but must relate the results back to the

model.

This area of modelling is an important feature of a CAS-enabled curriculum as,

with the ability of CAS to calculate, differentiate and integrate with much more

complex functions than would be practical without, it is possible to explore more

realistic questions. Consequently, students can now be asked modelling

questions which don’t feel as contrived as many of the modelling questions that

form part of the non-CAS examinations. An example of this can be seen in the

MEI Core 4 paper from January 2007 shown in figure 60.


Figure 61 - MEI Core 4 - January 2007

Notice how the description at the top of the question is never really referred to

again within the question. In this case, the question would have worked just as

well without the context supplied; this cannot be said for the example shown

earlier from the VCAA exam paper (on page 110).

From this quick exploration of questions from CAS-enabled examination papers

we can see a number of recurring themes: the use of a modelling context, the

use of additional parameters, step-wise functions and restrictions on the domain

of a function. Using these techniques, it is possible, as can be seen above, to


ask questions which are similar in style and content to those currently used in

non-CAS exams, but with an added layer of complexity. It is my opinion that

answering such questions demonstrates at least the same level of

understanding of the concepts behind the questions as answering their non-

CAS equivalents, even if some of the computational work is being carried out by

a CAS calculator. Some evidence to support this claim can be seen in Norton

et al (2007 pp. 544-549) but a more focused study would be necessary to fully

explore this conjecture.

I would argue that any student who could successfully answer the questions

above would have demonstrated, at the very least, the same level of

understanding of the concepts involved as someone who had completed a

standard A-level course and I would have no concern about them going on to

study Mathematics or a mathematically related degree course.

These questions for the most part demonstrate how CAS could be used to

augment our current curriculum, taking topics and concepts that are currently

studied, changing the emphasis from mechanical skills to interpretative skills

and extending those concepts by requiring an appreciation for more context

related problems, involving additional parameters or sophisticated modelling

contexts.

Whilst there is much that can and has been said about the pedagogical uses of

CAS within the classroom without any changes to our assessment system, I

think it is essential that we also consider how assessment within a CAS

curriculum could work. There is value in exploring what a completely new CAS


curriculum, designed from the ground up, might look like, in the same way as

Papert (1996) did in his exploration of a LOGO curriculum. However, it is

equally important to remember that nearly all curriculum changes are

‘evolutionary’ in nature, not ‘revolutionary’.

As McMullin (2003 p329) points out, whilst there are various possible methods

of assessment, the traditional exam, with short-answer multiple-choice

questions and more extended answers, is not going to disappear. Even if we

do not use timed written assessment in the classroom, external assessments

will take this form for some time to come. Whether we like it or not, there is a

degree to which “Assessment drives curriculum.” (2003 p329)

Once CAS is in the assessment it will quickly follow into the classroom.

Whether teachers approve of the use of CAS or not, once “national tests require

the use of technology and computer algebra systems, then teachers will teach

their students to use them” (McMullin 2003 p.329).

Once CAS has become widely used in classrooms and exams, it will then begin

naturally to find its place in syllabi. Until students are using the technology

regularly and extensively within the classroom it will not be possible to truly see

if, as I suspect, students will be able to progress further up the ‘tree of

mathematics’, and what mathematics it is now possible teach within our

syllabus.

The problem with looking for a complete rewrite of the secondary school

mathematics syllabus is that it is such a change is likely never to happen. What


is more likely to succeed is slow and gentle introduction of the technology into

our current curriculum and assessment alongside evolutionary changes over a

period of many years as we realise that some things that were once central are

no longer important as they once were, and things that were once beyond our

students are now accessible.

“Many things need to change to accommodate CAS. This necessity

is a good thing. In mathematics, CAS can do many things better

than humans, but we cannot simply hand students CAS and let them

go. Only when CAS use by students is required on traditional

assessments at the local, state and national level will CAS truly

become part of the curriculum - a change that must occur. Done

carefully and purposefully, traditional assessment can lead the way

to the inclusion of the use of this important new technology in the

high school mathematics curriculum and increase the amount of real

mathematics students can do” (McMullin 2003 p334 emphasis

added)

Bibliography 125

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